New polynomial and multidimensional extensions of classical partition results

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• 作者：Vitaly Bergelson¹ ; John H. Johnson Jr. ; Joel Moreira ^{joel.moreira@northwestern.edu}
• 关键词：Rado's Theorem ; Partition regularity ; Deuber system
• 刊名：Journal of Combinatorial Theory, Series A
• 出版年：2017
• 出版时间：April 2017
• 年：2017
• 卷：147
• 期：Complete
• 页码：119-154
• 全文大小：656 K
• 卷排序：147

文摘

In the 1970s Deuber introduced the notion of (m,p,c)(m,p,c)-sets in NN and showed that these sets are partition regular and contain all linear partition regular configurations in NN. In this paper we obtain enhancements and extensions of classical results on (m,p,c)(m,p,c)-sets in two directions. First, we show, with the help of ultrafilter techniques, that Deuber's results extend to polynomial configurations in abelian groups. In particular, we obtain new partition regular polynomial configurations in ZdZd. Second, we give two proofs of a generalization of Deuber's results to general commutative semigroups.We also obtain a polynomial version of the central sets theorem of Furstenberg, extend the theory of (m,p,c)(m,p,c)-systems of Deuber, Hindman and Lefmann and generalize a classical theorem of Rado regarding partition regularity of linear systems of equations over NN to commutative semigroups.